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Unformatted text preview: EE221A Linear System Theory Problem Set 1 Professor C. Tomlin Department of Electrical Engineering and Computer Sciences, UC Berkeley Fall 2011 Issued 9/1; Due 9/8 Problem 1: Functions. Consider f : R 3 R 3 , defined as f ( x ) = Ax,A = 1 1 1 1 ,x R 3 Is f a function? Is it injective? Is it surjective? Justify your answers. Problem 2: Fields. (a) Use the axioms of the field to show that, in any field, the additive identity and the multiplicative identity are unique. (b) Is GL n , the set of all n n nonsingular matrices, a field? Justify your answer. Problem 3: Vector Spaces. (a) Show that ( R n , R ), the set of all ordered n-tuples of elements from the field of real numbers R , is a vector space. (b) Show that the set of all polynomials in s of degree k or less with real coefficients is a vector space over the field R . Find a basis. What is the dimension of the vector space?...
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